Spectrality of a class of Moran measures on with consecutive digit sets

Abstract

Let $\{R_{k}\}_{k=1}^{\infty }$ be a sequence of expanding integer matrices in $M_{n}(\mathbb {Z})$ , and let $\{D_{k}\}_{k=1}^{\infty }$ be a sequence of finite digit sets with integer vectors in ${\mathbb Z}^{n}$ . In this paper, we prove that under certain conditions in terms of $(R_{k},D_{k})$ for $k\ge 1$ , the Moran measure $$ \begin{align*} \mu_{\{R_{k}\},\{D_{k}\}}:=\delta_{R_{1}^{-1}D_{1}}\ast\delta_{R_{1}^{-1}R_{2}^{-1}D_{2}}\ast\cdots \end{align*} $$ is a spectral measure. For the converse, we get a necessity condition for the admissible pair $(R,D)$ .